First-passage process in degree space for the time-dependent Erd\H{o}s-R\'enyi and Watts-Strogatz models

Abstract

In this work, we investigate the temporal evolution of the degree of a given vertex in a network by mapping the dynamics into a random walk problem in degree space. We analyze when the degree approximates a pre-established value through a parallel with the first-passage problem of random walks. The method is illustrated on the time-dependent versions of the Erd\H{o}s-R\'enyi and Watts-Strogatz models, which originally were formulated as static networks. We have succeeded in obtaining an analytic form for the first and the second moments of the first-passage time and showing how they depend on the size of the network. The dominant contribution for large networks with $N$ vertices indicates that these quantities scale on the ratio $N/p$, where $p$ is the linking probability.